# How do you find the domain & range for y=5cos(1/2)(x-(pi/4) +5?

##### 1 Answer
Jun 7, 2016

The domain of a sinusodial function is all the real numbers. The range, however, is determined by two things: the amplitude and the vertical displacement.

#### Explanation:

In function $f \left(x\right) = a \cos b \left(x - c\right) + d$, the amplitude is given by $| a |$ and the vertical displacement is given by $d$.

Therefore, your function has a vertical displacement of $5$ and an amplitude of $5$. A vertical displacement means you moved the graph up by $d$ units from the x axis. The amplitude is the distance between the center line ($y = 5$ in this case) and the maximum/minimum points of the function. Thus, the maximum points will be at $\left(x , 10\right) \mathmr{and} \left(x , 0\right)$. The can be summarized as $\text{minimum in y"<= y <= "maximum in y }$, or in the case of this function, $0 \le y \le 10$.

In conclusion, the domain of $y = 5 \cos \left(\left(\frac{1}{2}\right) \left(x - \frac{\pi}{4}\right)\right) + 5$ is $x = \text{all the real numbers}$ and the range is $0 \le y \le 10$.

Hopefully this helps!